On 12/12/2012 2:16 AM, William Elliot wrote:> On Tue, 11 Dec 2012, fom wrote:>> On 12/11/2012 9:58 PM, David R Tribble wrote:>>>> We see the phrase "almost infinite" (or "nearly infinite", or>>> "infinite for all practical purposes") in much literature for the>>> layman, usually to describe a vastly large number of combinations or>>> possibilities from a relatively large number of items. For example,>>> all of the possible brain states for a human brain (comprising about 3>>> billion neurons), or all possible combinations of a million Lego>>> blocks, etc.>>>>>> Obviously, these are in actuality just large finite numbers; having an>>> infinite number of permutations of a set of objects would require the>>> set to be infinite itself, or the number of possible states of each>>> element would have to be infinite. Most uses of the term "infinite>>> possibilities" or "almost infinite" are, in fact, just large finite>>> numbers. All of which are, of course, less than infinity.>>>>>> But is there some mathematically meaningful definition of "almost>>> infinite"? If we say that m is a "nearly infinite" number, where m <>>> omega, but with m having some property that in general makes it larger>>> than "almost all" finite n?>>>>>> Personally, I don't think there is such a definition; but then I would>>> enjoy being proved wrong.>>>> When people look at completeness of the real number system, they are>> looking at convergent sequences attaining a bound within some finite>> distance of the origin.>>> ... sequences attaining a bound within> some arbitrary distance of the limit.>>> The construction of the real numbers>> usually involves considering such sequences>> of rational numbers AS the real number.>> ... involves considering equivalence classes of such> sequences of rational numbers as the real numbers.>>> The "opposite" of this would be a divergent>> sequence that is unbounded.>> The opposite of a convergent sequence, by definition,> is a divergence sequence. A divergent sequence can> approach oo, -oo or oscillates or oscillates within a region.>>> The sequence itself never gets to and infinite distance from the origin.>> But, it grows larger than any convergent sequence.>>> The divergent sequence (sin n)_n doesn't grow larger than the convergence> sequence (3 - 1/n)_n.>>> Hopefully, someone may have a better suggestion for you.>> How so? It seems unrelated to his puzzling.>

As I was not concerned with making a statementso precise as to be incomprehensible to someoneasking a question about naive intuition, I was notmaking a claim about every sequence that does notconverge.

You know just enough mathematics to demonstratethe poverty of your personality.

As I recall, when we first met, you wereunable to recognize a properly negated conditionalin a proof.